Representation-Compatible Power Indices

TL;DR

This paper introduces representation-compatible power indices based on average representations in weighted games, addressing limitations of classical indices by ensuring coherence with voter equivalence classes and the distribution of weights.

Contribution

It proposes new power indices that are representation-compatible and can reveal voter equivalence classes, improving upon classical power indices.

Findings

Power indices are proportional to weight distributions in average representations.

New indices ascribe equal power to voters in the same equivalence class.

Addresses lack of coherence in classical power indices.

Abstract

This paper studies power indices based on average representations of a weighted game. If restricted to account for the lack of power of dummy voters, average representations become coherent measures of voting power, with power distributions being proportional to the distribution of weights in the average representation. This makes these indices representation-compatible, a property not fulfilled by classical power indices. Average representations can be tailored to reveal the equivalence classes of voters defined by the Isbell desirability relation, which leads to a pair of new power indices that ascribes equal power to all members of an equivalence class.